Analysis of non-unique solutions in mean field games

Livesay, Michael R.

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https://hdl.handle.net/2142/101031 Copy

Description

Title

Analysis of non-unique solutions in mean field games

Author(s)

Livesay, Michael R.

Issue Date

2018-04-19

Director of Research (if dissertation) or Advisor (if thesis)

Hajek, Bruce

Department of Study

Electrical & Computer Eng

Discipline

Electrical & Computer Engr

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

M.S.

Degree Level

Thesis

Keyword(s)

Mean field games, initial-terminal value problem, non-uniqueness

Abstract

This thesis investigates cases when solutions to a mean field game (MFG) are non-unique. The symmetric Markov perfect information N-player game is considered and restricted to finite states and continuous time. The players' transitions are random with a parameter determined by their control. There is a unique joint distribution of the players for the symmetric Markov perfect equilibrium, but there can be multiple solutions to the MFG equations. This thesis focuses on understanding the behaviors of the many MFG solutions for the 2-state case. This thesis explores methods to determine which MFG solution represents the fluid limit trajectories of the N-player system for large populations. This thesis investigates the MFG map which acts on the MFG distributions and outputs a prediction of the population's distribution based on the expected response of any given player. The MFG solutions are exactly the fixed points of the MFG map. The MFG solution that approximates large population trajectories is conjectured to be the only stable point for the MFG map. There is a second concept investigated, social cost, which is the average accumulated cost per player. But as is shown, the social cost is not a good indicator of which MFG solution approximates large population trajectories. A set, called the bifurcation set, is defined by there being some possibility of multiple trajectories of a large population. Another important set is the indifference set, which indicates when the transition rate of the players to a state is positively reinforced by an increase of the empirical distribution of that state. However, numerical results are given, indicating that the fluid limit trajectory may relate to stability of the MFG map. It appears the MFG map is difficult to handle in many ways; stability of the mapping is difficult to show, even in a simple example and there are numerical anomalies such that non-fixed points appear to be numerically stable under rigorous tests.

Graduation Semester

2018-05

Type of Resource

text

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http://hdl.handle.net/2142/101031

Copyright and License Information

Copyright 2018 Michael Richard Livesay

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